Subset Sums

Find the last \(3\) digits of the number of subsets (summing to \(1000\)) of the set of positive integers from \(1\) to \(100\) (both inclusive).

For example, the set:

\( \lbrace 1, 16, 30, 43, 47, 52, 53, 55, 57, 60, 64, 67, 68, 69, 71, 72, 75, 100 \rbrace \)

sums to \(1000\) and it is a subset of the set of positive integers from \(1\) to \(100\) (inclusive), so it would be included.

Subsets do not have ordered elements and a subset can not contain repeats of elements.

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